Here is a tutorial on how to make a stuffed animal, in this case an elephant, but you could follow the same steps with any kind of animal you want.
1st, draw out your pattern. I made mine on the computer and used a print as inspiration for the shape of the elephant. Just use the basic body shape and then make a copy of the lower half of the body. Cut out 2 of the body, 2 of the belly and 4 or the ears. Make sure you have two layers of material, with the backs facing together, when you cut them out. This way you will have two opposite pieces of material and you only have to cut out each shape once.
I poked a hole where the eye was supposed to go and marked it with a marker on the material, so I would know where to sew on the eye later.
Trace out your pattern on the material, you may need to enlarge the photo to see the lines.
Pin the two layers of material together and cut out your shapes.
Making the ears and the belly/legs, put the right sides of the material together, sew around the ears, leaving the tops open and sew along the top/center of the belly. (In hind sight, I would not have made the elephant's belly so rounded, it turned out cute, but the elephant looks pregnant :)
Iron the seam on the legs flat. Turn the ears inside out and top-stitch 1/8 inch in around the edge. (If you want, you could stuff the ears to make them more substantial)
Pin the ears in place, make sure they are the same one both sides of the elephant.
Sew the ears on only along the top of the ear, this way they flap.
Pin the legs to the body with the right sides facing each other. (Note: If you look closely, you will see that I have the elephant body right side facing out, I had to rip out my stitches and redo this step, don't do what is show in the picture!) Sew 1/4 inch in, from the center seam of the belly on one end of the elephant to the center seam on the other end of the elephant. DO NOT go past the center seam, even one stitch! Once you have the legs sewn attached on one side, sew on the other side of legs to the other half of the elephant body.
Make the tail. I used 3 strands of embroidery floss, braided together and tied at both ends, you can use yarn.
Place the tail, remember you will be turning the elephant inside out when it is finished, so you need to place the tail in the opposite direction of where you want it to end up. I wanted the tail to angle up a little, so I angled it down when placing it.
Lay the two pieces of elephant body flat and pin around the elephant body. Sew from one end of the elephant around to the other end. Make sure you do not sew through the leg material, even one stitch, it will pull the material up funny when you try and turn it inside out. This is a difficult start and stop stitching part, but just go slow and be precise. Your stitches need to meet up perfectly, but not overlap.
Now, once we get to this point, you may wonder how we are going to turn the elephant inside out, because we have completely sewn around the entire elephant! I opened up the belly by double stitching along the belly the seam about 1 inch back from the front of the elephant and then again about 3 inches back from there. I then ripped out the seam in between where I double stitched. Why? To keep the seam from coming apart. Why not just do this to begin with? I think it made it easier to sew the pieces together, having a solid stitch through the two belly pieces.
Ok, turn the elephant inside out, this is the fun part! It helps if you push out from the inside along the seams, to flatten them out.
Stuff the elephant, it is up to you how 'stuffed' you want to make your elephant, you can really pack in the stuffing, or leave it a bit loose for a squishier elephant. Once you have it stuffed, you need to put the eye on. This is where making the spot for the eye ahead of time is really helpful. Thread a needle, get the buttons you want to use for your eye, and use the dots you marked as a guide for where to sew the eyes on. Pull the thread through the elephant's head and place another button on the other side, so you sew on both eyes at the same time, threading back and forth, from side to side several times. You also need to sew the belly closed. You can do this however you want, and don't worry if it is not perfect, this is a handmade gift and that is a detail that will show that it is!
Finished elephant, left side
Finished elephant, right side, notice the pregnant belly :)